Optimal. Leaf size=107 \[ -\frac{a^2}{4 b^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a}{3 b^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0478855, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {646, 43} \[ -\frac{a^2}{4 b^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a}{3 b^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{x^2}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac{a^2}{b^7 (a+b x)^5}-\frac{2 a}{b^7 (a+b x)^4}+\frac{1}{b^7 (a+b x)^3}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{a^2}{4 b^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a}{3 b^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0135012, size = 44, normalized size = 0.41 \[ \frac{-a^2-4 a b x-6 b^2 x^2}{12 b^3 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.177, size = 37, normalized size = 0.4 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( 6\,{b}^{2}{x}^{2}+4\,abx+{a}^{2} \right ) }{12\,{b}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18181, size = 77, normalized size = 0.72 \begin{align*} -\frac{a^{2} b^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \, a b}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{1}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59905, size = 134, normalized size = 1.25 \begin{align*} -\frac{6 \, b^{2} x^{2} + 4 \, a b x + a^{2}}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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